I updated the paper (and/or we page) http://rangevoting.org/NewAppo.html with some more of the potpourri of apportionment methods you can get with different probabilistic models and different goals.
I'm actually now coming to the opinion that the alternative method #2 in the potpourri there, is actually the "morally best" of the lot. I might call this the "Ossipoff-Smith" method (at least were Ossipoff in the right mood for that, which he apparently is not), because the underlying theoretial attack is exactly that suggested by Mike Ossipoff for his "bias free Webster" method, except that the underlying probabilistic model is now an exponential distribtuion not a uniform "distribution" (I use the word in quotes since Ossipoff has in various ways ignored the requirements of probability theory, e.g. in his recent attack on the idea that probability distributions need to be normalizable; I will say that generally, when such an attack is required, it is a symptom of rot in one's underlying model which it would be better to fix). We get a more general formula than Ossipoff's that reduces to his in the limit K-->0 where K=#seats/#states=50/453. QUANTIFIED MORALITY That method aims to null out the "bias" which (for that method) is defined as the expected |additive change| in the number of seats a state has (caused by the round to integer) divided by state population. Why is this "morally right"? Well, it seems that the unfairness, for an individual person in that state, of having X times more congressmen than he ideally should, is |X-1|/StatePopulation. At first I thought that X=1/10 should be equally unfair as X=10 and hence was considering using unfairness formulas involving |log(X)|. However, I later thought X=10, X=100, etc keeps getting more and more unfair proportionally roughly to X, while X decreasing X=0.1, X=0.01 etc does not get much more unfair, and when X=0 it is essentially the same badness as X=0.1, whereas X=100 really is about 10 times worse than X=10. Now we have to weight this unfairness per person, by the number of people in the state, which means we multiply by StatePopulation thus cancelling out the denominator to get |X-1|. But that is exactly what the derivation of the method nulled out, because |X-1| is proportional to additive change in #seats divided by state population. OPEN QUESTIONS ABOUT THIS (AND THE OTHER METHODS IN http://rangevoting.org/NewAppo.html) Include whether we can now devise some pairwise optimality theorem or global optimality theorem that these methods obey (see http://rangevoting.org/Apportion.html for such theorems concerning the classic apportionment methods as well as an introduction to them generally). ------------------ Concerning Ossipoff's latest "AR" ("adjusted rounding") apportionment method, explained in his post titled "Detailed (but obvious) instructions for Adjusted-Rounding" it sounds interesting. To take a more abstract view of this: it seems to me that what you can accomplish with the idea of treating each "cycle" on its own, is to avoid having to use ANY probabilistic model. That's because the probability distribution within one "cycle," is just the data itself. We can now round the elements of that cycle, in such a way as to minimize our favorite bias measure. The sets that constitute cycles, however, depend on the global "divisor," which we have to choose to cause the total number of seats to come out right. It is not obvious (and requires proof) that, for any given bias-measure, the resulting method actually works, i.e. that a suitable divisor actually must exist. Also, it is not obvious (and requires proof) that two DIFFERENT divisors do not both exist that yield different apportionments. It seems to me that such proofs would follow from an appropriate general purpose lemma which says as you smoothly scale up the data within a "cycle" you get more seats one at a time (and never fewer, and never 2-hops), and if you add a new datapoint at the left endpoint n of the cycle, that increases the number of seats for that cycle by exactly n (or if add at right endpoint, n+1). Further this whole idea is not a "divisor method" in the sense of Balinski & Young et al, and hence ought to yield "monotonicity failures." Subjectively speaking, I do not see why the advantages that we can gain from going to this sort of method, are worth the cost of losing monotonicity (because such a loss seems based on the historical evidence to make a method politically unacceptable). Warren D Smith http://rangevoting.org ---- election-methods mailing list - see http://electorama.com/em for list info
